a	b	operation	Return type
scalar	scalar	Scalar multiplication	number
1D array	1D array	Vector dot product	number
2D array	1D array	Matrix-vector product	1D Numpy array
2D array	2D array	Matrix-matrix multiplication	2D Numpy array
n-D array	n-D array	Batch products	n-D Numpy array

Examples

Dotting two numbers (scalar and scalar)


        
        
            
                
                
                    np.dot(2, 3)
                
            
            6

WARNING

Avoid this for scalar multiplication. If you just want to multiply two scalar numbers, just use the standard 2 * 3 - it's faster and clearer.

Dotting two arrays (vector and vector)


        
        
            
                
                
                    # (1 * 3) + (2 * 4) = 11
np.dot([1,2], [3,4])
                
            
            11

Mathematically, we're doing the following:

$$\begin{pmatrix} 1\\ 3\\ \end{pmatrix} \cdot \begin{pmatrix} 2\\ 4\\ \end{pmatrix}= 11$$

Just as a side note, the parameters just have to be array-like; we can use NumPy arrays as well:


        
        
            
                
                
                    x = np.array([1,2])
y = np.array([3,4])
np.dot(x, y)
                
            
            11

Matrix-vector multiplication


        
        
            
                
                
                    X = np.array([[1,2],[3,4]])
y = np.array([5,6])
np.dot(X,y)
                
            
            array([17, 39])

Mathematically, we're doing the following:

$$\begin{pmatrix} 1&2\\ 3&4\\ \end{pmatrix} \begin{pmatrix} 5\\ 6\\ \end{pmatrix}= \begin{pmatrix} 17\\ 39\\ \end{pmatrix}$$

Matrix-matrix multiplication


        
        
            
                
                
                    x = [[1,0], [0,1]]
y = [[2,2], [2,2]]
np.dot(x, y)
                
            
            array([[2, 2],
       [2, 2]])

Mathematically, we're doing the following:

$$\begin{pmatrix} 1&0\\ 0&1\\ \end{pmatrix} \begin{pmatrix} 2&2\\ 2&2\\ \end{pmatrix}= \begin{pmatrix} 2&2\\ 2&2\\ \end{pmatrix}$$

Always remember that parameters just have to be array-like; we can use NumPy arrays as well:


        
        
            
                
                
                    x = np.array([[1,0], [0,1]])
y = np.array([[2,2], [2,2]])
np.dot(x, y)
                
            
            array([[2, 2],
       [2, 2]])

WARNING

Avoid this for matrix multiplication. If you want to take the product of two matrices, use NumPy's matmul(~) method or the @ notation instead.

Batch products

The dot(~) method can be used to compute multiple products at once, like follows:


        
        
            
                
                
                    x = [ [[1,0], [0,1]], [[1,1], [1,1]] ]
y = [3,4]
np.dot(x,y)
                
            
            array([[3, 4],
       [7, 7]])

In this example, the variable $x$ holds the following two matrices:

$$\begin{pmatrix} 1&0\\ 0&1\\ \end{pmatrix} \;\;\;\; \begin{pmatrix} 1&1\\ 1&1\\ \end{pmatrix}$$

The final line, np.dot(x,y), is performing the following mathematical operations:

$$\begin{pmatrix} 1&0\\ 0&1\\ \end{pmatrix} \begin{pmatrix} 3\\ 4\\ \end{pmatrix}= \begin{pmatrix} 3\\ 4\\ \end{pmatrix}$$

$$\begin{pmatrix} 1&1\\ 1&1\\ \end{pmatrix} \begin{pmatrix} 3\\ 4\\ \end{pmatrix}= \begin{pmatrix} 7\\ 7\\ \end{pmatrix}$$

Note that batch products also for vector-vector product and matrix-matrix product as well.

Published by Isshin Inada

Edited by 0 others

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